Question

решите уравнение пж sin^2(x/2)+sin^2(5x/2)+sin^2(2x)=2 срочно

Answer 1

$sin^2 \dfrac{x}{2}+sin^2 \dfrac{5x}{2} +sin^22x=2 \\ \dfrac{1-cosx}{2}+1-cos^2 \dfrac{5x}{2}+ \dfrac{1-cos4x}{2}=2 \\ 1-cosx-2cos^2 \dfrac{5x}{2}+1-cos4x=2 \\ 2cos^2 \dfrac{5x}{2}+cos4x+cosx=0 \\ 2cos^2 \dfrac{5x}{2}+2cos \dfrac{5x}{2}cos \dfrac{3x}{2}=0 \\ cos \dfrac{5x}{2}(cos \dfrac{5x}{2}+cos \dfrac{3x}{2})=0 \\ cos \dfrac{5x}{2}cos \dfrac{x}{2}cos2x=0 \\ \\ cos \dfrac{5x}{2}=0 \\ \dfrac{5x}{2}= \dfrac{\pi}{2}+\pi k \\ \boxed{x= \dfrac{\pi}{5}+ \dfrac{2\pi k}{5}; \ k \in Z}$

$cos \dfrac{x}{2}=0 \\ \dfrac{x}{2}= \dfrac{\pi}{2}+\pi k \\ \boxed{x= \pi +2\pi k ; \ k \in Z} \\ \\ cos2x=0 \\ 2x= \dfrac{\pi}{2}+ \pi k \\ \boxed{x= \dfrac{\pi}{4}+ \dfrac{\pi k}{2}; \ k \in Z }$